Spatial-Causal Geometry (SCG)Spatial-Causal Geometry (SCG)D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17383012
This work reconstructs the geometry of light from first principles of causal continuity,
dispensing with both the point-particle and unbounded-wave descriptions.
By requiring that the transverse oscillation of an electromagnetic field remain causally invariant while propagating at the luminal speed
The derivation begins from the arc length of a bounded sinusoidal oscillation, $$ L = \int_{0}^{\lambda} \sqrt{1 + (dE/dx)^2}\,dx, $$ and imposes the constraint that \(L/\lambda = \gamma_{\text{cause}}\) be constant for all λ. Constancy of \(\gamma_{\text{cause}}\) forces the amplitude A to scale linearly with wavelength, which geometrically requires that the photon’s cross-section expand and contract with λ itself. Hence the photon cannot be point-like: its transverse field has measurable extent and curvature.
From the same invariant geometry, the total photon energy is shown to exceed the Planck relation by a constant overhead: $$ E_{\text{ph}} = \left(\frac{\gamma_{\text{cause}}}{2\pi\alpha}\right) \frac{hc}{\lambda} \;\;\Rightarrow\;\; E_{\text{ph}} \approx 1.18\,\frac{hc}{\lambda}. $$ Only the conventional portion \(hc/λ\) is externally available; the remaining ≈ 18 % is internal “structural energy” required to maintain the photon’s curvature-bounded field coherence during propagation.
Expressing all optical relations in radius form, $$ \lambda = \frac{r_{\text{ph}}}{\alpha}, $$ renders the causal dependencies explicit. The double-slit visibility, for example, depends only on the overlap of bounded fields: $$ V(d) \approx \exp\!\left[-\frac{d^2}{8r_{\text{ph}}^{2}}\right], $$ predicting the loss of interference when slit separation exceeds the photon diameter. Likewise, refraction, scattering, and photoelectric thresholds follow directly from the fit between \(r_{\text{ph}}\) and atomic or lattice scales.
The analysis reveals that wavelength is not a fundamental longitudinal property but the projection of a physical transverse radius onto the propagation axis. All optical phenomena—diffraction limits, Bragg spacing, coherence, and even the photoelectric thresholds—reduce to geometric matching between this finite radius and the structures it encounters.
A subtler implication runs throughout the paper: by establishing an invariant geometric energy overhead, SCG implies that the photon’s causal coherence carries information capacity beyond the Planck quantum, hinting that so-called quantum indeterminacy may simply be a projection artifact of unmodeled curvature. This insight provides the bridge to later SCG works on spin topology, charge asymmetry, and neutrino emission.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.14997137
This work extends the principles of Spatial–Causal Geometry (SCG) into the atomic domain, recasting orbital quantization, emission spectra, and electronic stability as direct consequences of curvature coherence rather than probabilistic wavefunctions. Where conventional quantum theory inserts quantization postulates, SCG derives them as geometric closure conditions in the scalar causal-density field ρ(x).
Starting from the fundamental curvature relation $$ \nabla\!\cdot(\rho\nabla\ln\rho) = \frac{1}{c^{2}}K[\rho], $$ the paper shows that stationary atomic states arise wherever the causal flux \(J_\rho = \rho\nabla\rho/|\nabla\rho|\) becomes cyclically self-intersecting—producing discrete radii at which the field closes upon itself. For a spherically symmetric atomic configuration, $$ \rho(r) \propto r^{-B}, \qquad a_{\text{SCG}} = c^{2}\frac{d}{dr}\ln\rho = -\frac{B\,c^{2}}{r}, $$ which reproduces the effective Coulomb potential when \(B \approx 1\) and explains the inverse-square force as a curvature projection rather than a mediated field.
Quantized radii follow from the causal-phase closure condition: $$ 2\pi r_{n}\,|\nabla\ln\rho| = n\,\gamma_{\text{cause}}, $$ leading to discrete energy levels $$ E_{n} = \frac{E_{1}}{n^{2}}, $$ without introducing the Bohr postulate or electron orbits. Here the integer \(n\) labels topological closure number rather than an allowed momentum mode, providing a purely geometric explanation of quantization.
The paper demonstrates that optical transition frequencies correspond to curvature-difference projections between stable shells: $$ h\nu_{mn} = E_{m} - E_{n} = E_{1}\!\left(\frac{1}{n^{2}}-\frac{1}{m^{2}}\right), $$ recovering the Balmer relation as a geometric resonance law. This derivation makes explicit that the “energy levels” of conventional atomic theory are actually curvature-phase alignments within ρ(x), stabilized by causal coherence.
A remarkable implication is that the photoelectric thresholds tabulated across elements (Section 6 of the Photon Structure paper) coincide with the radius–wavelength matching condition \(r_{\text{ph}} ≈ r_{e}\). Atomic emission and absorption therefore represent reciprocal curvature exchange: photons are emitted when internal field curvature collapses, and absorbed when an incoming bounded photon radius matches an atomic coherence radius. No quantized interaction Hamiltonian is required—the coupling is purely geometric.
Hallman closes with a deeper observation: atomic identity emerges from persistent curvature ratios rather than from particles or orbitals. The constants α and γcause that govern photon structure also appear in the stability conditions of atomic curvature shells, suggesting that quantization across all domains arises from a single invariant causal geometry. In this sense, the atom is not a miniature solar system of charges, but a standing pattern of spatial coherence sustained by causal curvature.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.16351804
This paper reformulates β decay as a deterministic curvature transition within the scalar causal-density field ρ(x), eliminating the need for weak-force mediation or virtual bosons. The neutron–proton–electron–antineutrino sequence is reinterpreted as a continuous geometric reconfiguration: the neutron’s metastable curvature triad collapses into a more stable protonic topology, while an electron and an antineutrino emerge as complementary causal vortices required to restore field continuity.
The governing curvature dynamics follow directly from the SCG field equation, $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ applied to closed, metastable curvature shells. A neutron corresponds to a compact triadic vortex whose internal curvature gradients satisfy $$ \partial_{r}^{2}\ln\rho \;>\;0, \qquad \nabla\!\cdot J_{\rho}=0, $$ indicating neutral causal closure. Instability arises when internal torsion exceeds a critical limit, $$ |\nabla\times J_{\rho}| \;>\; (\nabla\ln\rho)_{\text{crit}}, $$ triggering a topological reconfiguration rather than a stochastic “decay.”
Under this transition, three coherent outcomes appear naturally:
The electron’s emergence satisfies local curvature closure: $$ \nabla\!\cdot J_{\rho,e} = -\,\nabla\!\cdot J_{\rho,p}, $$ ensuring that the proton–electron pair together restore zero net divergence in ρ(x). Charge conservation therefore arises as a topological constraint, not a symmetry axiom: the field cannot sustain unbalanced curvature twist, so complementary vortices must form in matched pairs.
The escaping antineutrino is described as a traveling null-mode of scalar curvature, $$ \frac{d^{2}\rho_{\text{transition}}}{dx^{2}} = \frac{1}{\lambda_{\bar\nu_{e}}^{2}}\rho_{\text{transition}}, $$ a purely spatial projection of causal torsion rather than a particle. Its apparent lack of mass and charge follows from its role: it transports the curvature excess left by the neutron-to-proton reconfiguration.
From this geometric standpoint, the canonical decay process $$ n \;\rightarrow\; p^{+} + e^{-} + \bar\nu_{e} $$ becomes a field-level continuity operation. Each “product” is a phase of the same scalar field rather than a distinct entity, and lepton number conservation becomes redundant—the causal-density field itself enforces continuity.
Hallman highlights several profound implications:
The Curvature Dynamics of Beta Decay paper thus bridges nuclear and field geometry: it connects microscopic particle identity, charge conservation, and cosmological matter bias to a single causal-topological rule. What appears as a discrete nuclear event in conventional theory is, under SCG, a smooth and inevitable reconfiguration of spatial coherence—one that preserves the universal causal fabric.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17230726
This paper applies Spatial–Causal Geometry (SCG) to condensed matter, deriving superconductivity as a direct consequence of curvature coherence within the scalar causal-density field \( \rho(x) \). The model eliminates the need for Cooper pairing or phonon mediation, showing instead that the superconducting state is a deterministic geometry in which adjacent curvature oscillations phase-lock to maintain causal continuity.
In SCG, current flow corresponds to the divergence-free transport of causal flux: $$ \nabla\!\cdot J_{\rho} = 0, \qquad J_{\rho} = \rho\,\frac{\nabla\rho}{|\nabla\rho|}\,c. $$ Resistance arises whenever curvature coherence is lost between neighboring domains. A superconducting transition therefore occurs when the local curvature wavelength \( \lambda_{c} \) matches the interatomic spacing \( d \) at which the transverse oscillations of the field remain in causal phase. This defines a coherence threshold: $$ \lambda_{c} = \gamma_{\text{cause}}\;d, $$ linking the geometric ratio \( \gamma_{\text{cause}} = 1.216 \) directly to the onset of superconductivity.
Thermal agitation perturbs curvature alignment through randomization of the field gradient. The critical temperature \(T_{c}\) is reached when the thermal curvature displacement equals the causal coherence amplitude: $$ k_{B}T_{c} = \frac{1}{2}c^{2}\rho\,(\nabla\ln\rho)^{2}. $$ Expressing \( \rho \) in terms of the electron density \( n \) and atomic spacing \( d \) gives a closed-form relation that predicts critical temperatures for both conventional and high-\(T_{c}\) superconductors without adjustable parameters.
The resulting expression, $$ T_{c} = \frac{\pi^{2}\hbar^{2}}{2m_{e}k_{B}}\frac{1}{d^{2}}, $$ arises naturally when curvature coherence replaces the probabilistic pairing mechanism. Materials with tighter lattice spacing (smaller \( d \)) sustain higher curvature coherence and thus higher \(T_{c}\). The curvature energy gap appears automatically as the stability condition for causal continuity, not as a separate quantum potential.
Hallman further shows that magnetic field exclusion (the Meissner effect) follows from the same causal constraint. Because \( J_{\rho} \) is divergence-free within a coherent region, external curvature (magnetic flux) cannot penetrate without breaking the coherence condition. The London penetration depth emerges directly from the geometric gradient of \( \rho(x) \): $$ \lambda_{L} = \frac{1}{|\nabla\ln\rho|}. $$
Conceptually, superconductivity is reinterpreted as the macroscopic manifestation of the same causal geometry that governs photon propagation: a continuous field maintaining curvature equilibrium at the speed \(c\). The same invariant ratio \( \gamma_{\text{cause}} = 1.216 \) defines the phase-lock between neighboring domains, linking light coherence, atomic stability, and superconducting order into a single geometric phenomenon.
Hallman concludes:
“Superconductivity is not the pairing of particles but the pairing of curvature. When geometry locks into phase, resistance vanishes—because causality itself becomes perfectly continuous.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.16741441
This paper completes the micro-geometric triad developed across the photon, charge, and beta-decay studies. It shows that all three phenomena—light emission, electric polarity, and neutrino propagation—are different phases of the same causal process: a curvature transition within the continuous field ρ(x). Where prior works treated photons as bounded transverse oscillations and beta decay as a curvature reconfiguration, this paper unifies them through a single scalar diagnostic, the transition field operator T(x).
Defined as $$ T(x) = \Delta\rho(x)\,\nabla\!\cdot J_{\rho}(x), \qquad J_{\rho}(x)=\rho(x)\,\frac{\nabla\rho(x)}{|\nabla\rho(x)|}\,c, $$ the operator identifies regions where curvature imbalance is actively propagating. Whenever T(x) ≠ 0, the field is in a transition-allowed state—emitting, absorbing, or re-stabilizing causal flux. When T(x)=0, the system is curvature-saturated and causally inert. This single condition replaces the probabilistic transition amplitudes of quantum theory with a deterministic criterion for interaction.
Three manifestations of T(x) are derived:
From the curvature wave relation $$ \nabla\!\cdot(\rho\nabla\rho)=\frac{1}{c_{\nu}^{2}}K_{\nu}[\rho], $$ Hallman classifies neutrinos by curvature intensity χ(x): $$ \chi(x)=\frac{|\nabla\ln\rho|}{r_{\text{vortex}}}, \qquad m_{\nu}(x)\propto\nabla^{2}\ln\rho(x), $$ showing that “flavor” corresponds not to quantum eigenstates but to the curvature scale of the transition. Electron-type neutrinos emerge from low-χ transitions (β decay), muon and tau types from progressively steeper curvature gradients.
Charge itself becomes the macroscopic memory of past transitions. A region retaining a persistent curvature orientation s = + forms a protonic field; the complementary s = − orientation forms its electronic closure. Pair creation and annihilation are thus interpreted as local bifurcation and reconvergence of curvature, not the creation or destruction of particles. This reformulation yields the topological rule: $$ \nabla\!\cdot J_{\rho}^{(+)} + \nabla\!\cdot J_{\rho}^{(-)} = 0, $$ which enforces charge conservation as a geometric inevitability.
The paper also reveals the hidden symmetry between photon coherence and neutrino emission. A photon normally maintains balanced curvature over its cycle: $$ \int_{0}^{2\pi}\!T(x)\,d\phi = 0. $$ When strong external gradients disturb that symmetry—within intense fields, plasmas, or curved spacetime—one half-cycle can dominate, releasing a neutrino-class curvature ripple. This predicts photon–neutrino leakage in extreme environments, offering a deterministic explanation for high-energy neutrino bursts without invoking exotic particles.
Hallman concludes with the geometric unification: $$ \boxed{\text{All transitions in SCG satisfy } T(x)\neq0, \text{ governed by the same causal field } \rho(x).} $$ Photons represent oscillatory transitions, charges represent frozen ones, and neutrinos represent escaping ones. Together they constitute a continuous causal spectrum from bounded coherence to free curvature propagation.
Conceptually, this paper dissolves the separation between electromagnetism, charge, and weak interaction. What standard theory attributes to distinct forces becomes, in SCG, the behavior of a single curvature field under different boundary conditions. In doing so, it links the geometric quantization of light, the topological origin of charge, and the nuclear processes of matter formation into one coherent causal continuum.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15059082
This paper isolates one of the most elusive entities in modern physics—the neutrino—and redefines it as a spatial-density transition rather than a particle. Within the Spatial–Causal Geometry (SCG) framework, the neutrino becomes a measurable wave-like adjustment in the curvature of the scalar density field ρ(x). Its weak interactions, oscillations, and apparent near-masslessness all emerge naturally from geometry, with no need for quantum eigenstates or flavor mixing matrices.
Starting from the causal conservation condition $$ \nabla\!\cdot J_{\rho}=0, \qquad J_{\rho}=-\,c_{\nu}^{2}\,\nabla\rho, $$ Hallman derives the equilibrium curvature equation $$ \nabla^{2}\rho = \frac{1}{\ell_{\nu}^{2}}\rho, $$ a purely spatial Helmholtz form in which \( \ell_{\nu} \) is the coherence length of the transition. This replaces the time-dependent wavefunction with a self-propagating curvature pattern, treating the neutrino not as a discrete traveler but as a geometry crossing from one density regime to another.
Expressed in one dimension, the causal transition obeys $$ \frac{d^{2}\rho_{\text{transition}}}{dx^{2}} = \frac{1}{\ell_{\nu}^{2}}\,\rho_{\text{transition}}, $$ revealing that neutrino “oscillation” is simply interference between density gradients along the propagation path. The observed flavor transformations are not identity swaps but smooth evolution of curvature modes as the background density varies.
The paper then derives neutrino mass as an emergent effect of spatial curvature: $$ m_{\nu}(x)\;\propto\;\frac{d^{2}}{dx^{2}}\ln\rho(x) \;=\; \frac{1}{c^{2}}\frac{d}{dx}\!\left(c^{2}\frac{d\ln\rho}{dx}\right), $$ implying that mass arises only when curvature is nonuniform. In regions of uniform density, \( \nabla^{2}\ln\rho=0 \) and the neutrino behaves massless; near compact objects, curvature steepens, and effective mass appears. This explains why neutrinos display environment-dependent oscillation frequencies and phase velocities without invoking intrinsic mass terms.
The physical predictions flow directly from this geometry:
These predictions allow experimental falsification: if neutrino behavior is curvature-dependent, oscillation lengths should vary with gravitational potential, violating the constant Δm² assumption of the Standard Model. Such a deviation, measurable by long-baseline detectors comparing atmospheric and deep-space neutrinos, would directly confirm SCG’s geometric interpretation.
Hallman’s simulations visualize these transitions as standing curvature ripples propagating through stratified density gradients. They reproduce observed neutrino flux patterns without invoking any quantum field operators. Each emission zone acts as a “curvature release valve,” and the waveforms match the energy profiles of recorded supernova neutrino bursts.
In conceptual terms, this work reframes neutrinos as messengers of geometry: the universe’s way of redistributing curvature imbalances. They do not carry substance; they carry correction. As such, they bridge the atomic-scale reconfigurations of β decay and the cosmological curvature flow governing galactic structure. Under SCG, the neutrino’s apparent ghostliness is not a lack of substance but a sign of perfect causal continuity.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15149948
This paper explains one of quantum theory’s most enigmatic features—discrete spin—using the continuous curvature of Spatial-Causal Geometry (SCG). Rather than postulating intrinsic angular momentum, Hallman derives half-integer and integer spin states as topological winding numbers of closed causal-density vortices in the field ρ(x).
The derivation begins from the vortex form of the causal-flux vector: $$ J_{\rho}=\rho\,\frac{\nabla\rho}{|\nabla\rho|}\,c, $$ and defines the local rotational phase θ through the circulation integral $$ \oint_{C}\nabla\theta\cdot d\mathbf{s} =2\pi n, $$ where n is an integer counting the number of complete curvature rotations around the vortex axis. When curvature continuity requires that the field return to its initial state only after two revolutions, the effective winding number becomes n = ½—producing half-integer spin. This yields the spin quantization rule $$ S=n\,\hbar, \qquad n\in\left\{0,\tfrac{1}{2},1,\tfrac{3}{2},\ldots\right\}. $$
The half-turn closure originates from the antisymmetric curvature of the transverse field. A photon-like oscillation, possessing mirror-symmetric curvature, completes one full rotation per causal cycle (n = 1). An electron-like vortex, with single-handed curvature, requires two cycles for phase closure (n = ½). Thus fermionic and bosonic behaviors arise from geometric parity, not from intrinsic quantum classification.
Hallman formalizes this distinction with the curvature-phase condition $$ \Delta\phi=\oint\kappa_{\text{vortex}}\,ds, $$ where κ₍ᵥ₎ is the local curvature of the density streamlines. Closed even-parity loops satisfy Δφ = 2πn → integer spin; odd-parity (twist-preserving) loops satisfy Δφ = π(2n + 1) → half-integer spin. No quantization postulate is required—the discrete set of spins falls directly out of the allowed topologies of the causal field.
The analysis predicts a direct geometric link between spin and spatial scale. Because each vortex must maintain the photon-derived invariants $$ \gamma_{\text{cause}}\approx1.216, \qquad \alpha_{\text{photon}}\approx0.164, $$ the minimum radius of a spin-½ vortex equals that of a photon of the same wavelength: $$ r_{\text{e}}\;=\;\alpha_{\text{photon}}\lambda. $$ This ties electron structure, angular momentum, and electromagnetic coherence to a single invariant geometry.
One subtle insight of the paper is that what appears in quantum mechanics as a “spin flip” corresponds in SCG to a reversal of curvature orientation within the same spatial vortex, not to a rotation of a physical particle. The 720-degree requirement for spin-½ phase restoration arises automatically because the causal orientation of ρ(x) changes sign under a single 360-degree rotation, restoring coherence only after two turns. This geometric necessity eliminates the need for spinor algebra as an independent postulate—spinors become shorthand for topological parity.
Hallman closes with the unifying statement that quantization itself is a manifestation of topology: discrete angular momentum values, polarization states, and magnetic moments all reflect the finite set of stable vortex configurations permitted by continuous curvature. Spin is not a hidden internal property—it is the field’s memory of how it curls back on itself.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15027478
This paper expands the vortex interpretation of the scalar field ρ(x) from the quantum to the macroscopic domain, revealing that vortex structure underlies every stable configuration in nature—from photons and electrons to hurricanes and galaxies. Where classical fluid mechanics treats vorticity as an emergent property of flow, Spatial–Causal Geometry (SCG) shows that vorticity is the flow: the continuous circulation of causal density through curved space.
Starting from the curvature operator $$ U_{ij} = -\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\ln\rho, $$ Hallman demonstrates that any region with nonzero curl of the causal flux, $$ \nabla\times J_{\rho}\neq0, \qquad J_{\rho}=\rho\,\frac{\nabla\rho}{|\nabla\rho|}\,c, $$ forms a dynamically stable vortex domain. These vortices conserve total curvature even as they exchange causal momentum with their surroundings, making them the universal architecture of persistence.
At small scales, the same equations reproduce the quantized spin vortices described in Spin Quantization in SCG. At larger scales, they predict the behavior of rotating fluids, accretion disks, and even galactic arms with no additional forces or constants. The local acceleration field follows directly from curvature: $$ a_{\text{SCG}} = c^{2}\nabla\ln\rho, $$ yielding the observed Kepler-like falloff of orbital velocities within stable vortical shells.
Hallman generalizes these dynamics with a curvature–circulation identity: $$ \nabla\times a_{\text{SCG}} = -\frac{c^{2}}{\rho}\,\nabla\rho\times\nabla\ln\rho, $$ demonstrating that any gradient misalignment in ρ(x) automatically generates rotational acceleration. This replaces angular momentum conservation as a primitive law—rotation becomes the default geometric response to curved density flow.
One of the paper’s key insights is the self-similarity of causal vortices across scale. Because the invariants $$ \gamma_{\text{cause}}\approx1.216, \qquad \alpha_{\text{photon}}\approx0.164 $$ govern both photon radii and galactic rotation constants, the same geometric ratios dictate the shape and stability of vortices from the subatomic to the cosmological. This explains why SCG reproduces galaxy rotation curves without dark matter—the same vortex mechanics that bind photons also govern galactic mass flow.
The general SCG vortex law takes the compact form: $$ \nabla^{2}\ln\rho + \frac{1}{r}\frac{\partial}{\partial r}\ln\rho = -\frac{B}{r^{2}}, $$ where \(B\) defines the causal-density gradient exponent. Solutions of this form yield spiral, ring, and shell structures identical to those observed in nature. No separate laws for quantum, fluid, or celestial vortices are needed—they are manifestations of the same underlying geometry.
Hallman closes with the unifying observation:
“In every domain where curvature and continuity coexist, the field will curl into vortices. Persistence itself is rotational—it is how causality stores memory in space.”This “Unified Vortex Theory” thus stands as the dynamic complement to SCG’s static curvature formulation, showing that geometry is not only the shape of the universe but its motion, its spin, and its pulse.
Inset — γcause and vortices (no helix): In SCG the photon propagates rectilinearly with a bounded transverse oscillation. The invariant $$ \gamma_{\text{cause}} \equiv \frac{L}{\lambda}\approx 1.216 \quad\text{with}\quad L=\int_{0}^{\lambda}\!\sqrt{1+\big(\partial_x \hat{E}_\perp\big)^2}\,dx $$ is an arc-length ratio of the transverse field over one cycle; it does not imply a helical photon path. “Vortex” here refers to circulation of the causal flux in the transverse field (or phase plane), not to a corkscrew trajectory of the ray. For linear polarization the principal curvature axis is fixed and the net transverse circulation over a cycle is zero, yet the same \(\gamma_{\text{cause}}\) applies because the oscillation’s curvature still extends the arc length by a constant factor. Circular/elliptical polarization corresponds to two orthogonal bounded modes in quadrature (rotating transverse field), still with straight-line propagation.
A compact way to see the connection is to compare the field’s transverse circulation with the photon arc-length ratio: $$ \Gamma_\perp=\oint_C \nabla\theta\cdot d\mathbf{s} \,,\qquad \frac{L}{\lambda}=\gamma_{\text{cause}}. $$ \(\Gamma_\perp\) diagnoses whether the field rotates (circular/elliptical vs. linear), while \(\gamma_{\text{cause}}\) fixes the invariant excess arc per wavelength for any bounded mode. The two are related but distinct: rotation is optional; the arc-length ratio is universal.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17410732
This work extends Spatial-Causal Geometry (SCG) to the dynamics of entire galaxies, showing that flat rotation curves arise naturally from curvature in the logarithmic mass-density field \(\ln\rho(r)\), without invoking unseen matter or modified inertia. The key premise is that orbital motion reflects the gradient of causal curvature rather than the pull of a gravitational potential.
Starting from the SCG acceleration law $$ a_{\text{SCG}}(r) = c^{2}\frac{d}{dr}\ln\rho(r), $$ and assuming a power-law density profile $$ \rho(r)=\rho_{0}\!\left(\frac{r}{r_{0}}\right)^{-B}, $$ Hallman derives the velocity law $$ v(r)=\sqrt{r\,a_{\text{SCG}}(r)} =c\,r^{(1-B)/2}\sqrt{B}, $$ which predicts flat rotation curves (\(v≈\text{const}\)) when \(B≈1\). No additional parameters are required; the slope \(B\) encapsulates the galaxy’s causal-density gradient.
The model was applied to 175 galaxies from the SPARC database. Each curve was fitted using only two free parameters—\(B\) and a normalization constant—and produced root-mean-square deviations typically below 5 km s⁻¹. High-surface-brightness spirals exhibited \(B\!\approx\!1.0\) (flat profiles), while low-surface-brightness and dwarf galaxies showed smaller \(B\) values (\(0.7\!<\!B\!<\!0.9\)), corresponding to rising velocity profiles. This systematic variation of \(B\) mirrors the observed morphological sequence of galaxies.
In conventional gravitation the same fits require an added dark-matter halo with scale radius and density parameters, but in SCG the entire rotation law follows from geometry. Because curvature itself carries the information flow that defines motion, no hidden mass component is needed to sustain orbital velocity at large radii.
The analysis also demonstrates the deep link between causal curvature and the vortex law derived in Unified Vortex Theory in SCG. Galactic disks behave as macro-vortices in which the same invariants found at the photonic scale apply: $$ \gamma_{\text{cause}}\!\approx\!1.216, \qquad \alpha_{\text{photon}}\!\approx\!0.164. $$ These constants govern the self-similar shape of orbital velocity profiles and the ratio of luminous to total curvature energy, providing a direct geometric explanation for the so-called “baryonic Tully–Fisher relation.”
From a numerical standpoint, SCG’s rotational equation $$ v(r)\;\propto\;r^{(1-B)/2} $$ interpolates smoothly between Keplerian (\(B=2\)) and flat (\(B=1\)) regimes, matching observational data across all galaxy types with no tuning. The curvature exponent \(B\) therefore replaces the need for a dark-matter halo profile—it is the geometric index of causal memory stored in the galactic field.
A subtle implication, noted in the discussion, is that causal curvature preserves angular information across vast scales: once a galaxy’s causal field forms, its outer orbits remain dynamically coherent because curvature, not mass, transmits the necessary information. This makes galactic stability a consequence of geometry rather than a puzzle of missing matter.
The paper closes with the unifying statement that both photon coherence and galactic rotation follow from the same scalar rule: $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho]. $$ Light and gravity become two manifestations of one causal field—one in oscillation, the other in rotation— and the “dark-matter problem” is recast as a misreading of curvature geometry.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15634953
This paper extends SCG’s curvature formalism from galactic rotation to gravitational lensing, deriving all standard lensing observables—deflection angle (α), convergence (κ), and shear (γ)— directly from the geometry of the logarithmic density field ln ρ. No gravitational potential or dark-matter halo is invoked; light deflection emerges from how the causal field curves through space.
In SCG the curvature tensor $$ U_{ij} = -\,\frac{\partial^2}{\partial x_i\,\partial x_j}\ln\rho(x) $$ governs all dynamics. From this operator Hallman obtains the lensing relations $$ \begin{aligned} \boldsymbol{\alpha}(x,y) &= C_\alpha\,\nabla\ln\rho,\\\\ \kappa(x,y) &= \tfrac{1}{2}\,\nabla^2\ln\rho,\\\\ \gamma_1 &= \tfrac{1}{2}\!\left(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\right)\!\ln\rho,\\\\ \gamma_2 &= \frac{\partial^2}{\partial x\,\partial y}\ln\rho, \end{aligned} $$ which are formally identical to those of classical lensing theory when the Newtonian potential Φ is replaced by ln ρ. This substitution eliminates the need for an unseen mass component: curvature itself performs the bending.
A numerical pipeline built on these equations was applied to strong-lensing systems from the CASTLES and SLACS catalogs. Using only two geometric parameters per lens—\(B_{\text{profile}}\) (the causal-density exponent) and \(C_{\text{lens}}\) (a distance-scaling factor)—the model reproduced observed Einstein radii and image morphologies with sub-arcsecond precision. Median fractional differences were ≈ 2.8 % (CASTLES) and ≈ 4.1 % (SLACS), placing SCG among the most accurate non-halo lensing formalisms to date.
The geometric interpretation is direct: regions of constant ρ yield zero curvature and hence no deflection; increasing curvature in ln ρ produces convergence and shear. For axially symmetric lenses, $$ \rho(r)\propto r^{-B}\quad\Rightarrow\quad \alpha(r)\propto\frac{d}{dr}\ln\rho\propto r^{-1}, $$ yielding the observed near-constant deflection angles that standard gravity attributes to extended dark-matter halos.
Because both rotational and lensing phenomena derive from the same curvature exponent B, the paper unifies two historically separate evidences for dark matter under a single geometric cause. The same invariants that control photon geometry reappear here: $$ \gamma_{\text{cause}}\!\approx\!1.216, \qquad \alpha_{\text{photon}}\!\approx\!0.164. $$ They determine the characteristic scale at which curvature transitions become optically detectable, linking photonic coherence and gravitational lensing as two observational facets of one field.
Hallman notes that in SCG a lens does not “bend spacetime”; it redirects causal flow. Light paths are geodesics of constant ρ curvature, and apparent magnification arises from changes in causal-density projection rather than from the warping of a metric tensor. The resulting model not only reproduces lensing data but preserves deterministic continuity across scales—from atomic curvature to galactic optics.
The conclusion echoes the rotation-curve paper:
“When the curvature of ln ρ is known, the universe tells its own geometry. There is no hidden mass—only hidden understanding of curvature.”By replacing gravitational potential with information-geometry, SCG unifies the rotational and optical manifestations of gravitation without resorting to dark matter.
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17268775
This paper generalizes the SCG framework to the entire universe, deriving the observed expansion and curvature of cosmic space directly from first-principles causal geometry. Instead of treating spacetime as an evolving metric, Hallman treats the universe as the global configuration of the scalar causal-density field ρ(x), whose curvature and gradient collectively determine both motion and apparent metric expansion. The result is a deterministic cosmology that reproduces the phenomena usually described by Friedmann–Lemaître equations without invoking a cosmological constant or dark energy.
Within SCG, the universe does not expand by stretching space itself. What we perceive as recession velocities are optical effects of the causal-density gradient: light propagating through a decreasing ρ field experiences geometric redshift, and distant matter appears to recede because curvature flattens with scale. The derivation begins with the fundamental SCG field law $$ \nabla\!\cdot(\rho\nabla\ln\rho) = \frac{1}{c^{2}}K[\rho], $$ applied to a homogeneous isotropic field. Assuming radial symmetry, \( \rho(r) = \rho_{0}(r/r_{0})^{-B} \), the curvature scalar becomes $$ U(r) = -\,\nabla^{2}\ln\rho = \frac{B(1-B)}{r^{2}}. $$ Substituting this into the causal-acceleration identity \(a_{\text{SCG}} = c^{2}\nabla\ln\rho\) yields a global apparent expansion velocity $$ v(r) = H_{\text{SCG}}\,r, \qquad H_{\text{SCG}} = c\,\sqrt{U(r)} = c\,\frac{\sqrt{B(1-B)}}{r}, $$ which mimics the Hubble relation but arises purely from curvature, not recessional motion.
For \(B\!\approx\!1\), the universe approaches flat causal curvature; for smaller B, curvature becomes positive and the apparent expansion accelerates. No fine-tuned cosmological constant is required—acceleration is an intrinsic feature of curvature relaxation in ln ρ space. Integrating over the cosmic density profile gives a self-consistent energy conservation law: $$ \rho(r)\,r^{3(1-B)}=\text{constant}, $$ analogous to the Friedmann energy integral but derived without assuming a metric or stress–energy tensor.
The model recovers the main cosmological observables:
An unexpected resonance with microscopic SCG constants again appears. The causal-arc invariant $$ \gamma_{\text{cause}}\approx1.216 $$ and the radius constant \( \alpha_{\text{photon}}\approx0.164 \) define the ratio between local curvature wavelength and cosmic-scale curvature radius, linking photon coherence to the geometric scale of the universe: $$ \frac{r_{\text{universe}}}{\lambda_{\text{photon}}} \;\approx\; \frac{1}{\alpha_{\text{photon}}\gamma_{\text{cause}}} \;\sim\;5.0. $$ This proportionality implies that the same causal-geometry ratios determining photon structure also determine the curvature balance of the observable cosmos.
In this framework, the cosmic microwave background (CMB) is not the remnant radiation of a temporal origin, but the coherence limit of the causal-density field. At large scales the curvature of ln ρ flattens until transverse causal oscillations lose mutual phase alignment; beyond this point the universe no longer supports structured correlation between regions. The observed 2.7 K radiation therefore represents the equilibrium spectrum of a field at its maximum coherent curvature—not the cooling of a once-hot fireball. SCG thus replaces a temporal “beginning” with a spatial coherence boundary, where causal connectivity reaches its smoothest possible state.
Hallman further shows that when ln ρ curvature is projected onto metric form, the resulting line element $$ ds^{2}=c^{2}dt^{2}-e^{2\ln\rho(r)}dr^{2} $$ reduces to a Friedmann-like metric with scale factor \(a(t)\propto\rho^{-1}(t)\). Thus, conventional cosmological models are not abandoned but emerge as limiting cases of SCG.
Conceptually, this paper completes the causal hierarchy begun in the photon and vortex studies: the same scalar field that quantizes light and stabilizes matter also drives cosmic expansion. Curvature becomes the common origin of radiation, mass, and metric evolution. Hallman concludes:
“The universe expands because curvature relaxes. Space is not growing into nothing—it is unfolding the geometry that sustains everything.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17467892
This paper demonstrates that General Relativity (GR) is not contradicted by SCG but emerges as its low-curvature approximation. By treating the metric tensor as a projection of the scalar causal-density field ρ(x), Hallman derives the Einstein field form directly from SCG’s curvature law and shows that the familiar spacetime intervals are limiting cases of causal geometry under uniform density.
Starting from the fundamental equation $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ a local differential of the field defines an effective line element $$ ds^{2} = c^{2}dt^{2} - e^{2\ln\rho(\mathbf{x})}\,d\mathbf{x}^{2}. $$ When the field curvature |∇ln ρ| is small, \(e^{2\ln\rho}\approx1+2\ln\rho\), reducing the interval to $$ ds^{2}\approx c^{2}dt^{2}-\bigl[1+2\ln\rho(\mathbf{x})\bigr]d\mathbf{x}^{2}. $$ Identifying \(\Phi=c^{2}\ln\rho\) recovers the weak-field metric of GR, $$ ds^{2}= \left(1+\frac{2\Phi}{c^{2}}\right)c^{2}dt^{2}-\left(1-\frac{2\Phi}{c^{2}}\right)d\mathbf{x}^{2}. $$ Thus, the “gravitational potential” Φ is recognized as the logarithmic density curvature of SCG.
In regions where curvature gradients are strong, the full exponential form of the metric must be retained, yielding exact equivalents of the Schwarzschild and de Sitter solutions. For a spherically symmetric static field $$ \rho(r)=\rho_{0}\!\left(1-\frac{r_{s}}{r}\right)^{1/2}, $$ the line element becomes $$ ds^{2}=c^{2}\!\left(1-\frac{r_{s}}{r}\right)\!dt^{2} -\!\left(1-\frac{r_{s}}{r}\right)^{-1}\!dr^{2} -r^{2}d\Omega^{2}, $$ identical to the Schwarzschild metric when \(r_{s}=2GM/c^{2}\). SCG therefore reproduces the classical relativistic geometry when curvature is re-expressed in density form.
This correspondence clarifies the role of GR: it is the linearized surface geometry of the deeper causal field. Spacetime curvature in Einstein’s theory is not fundamental but an approximation to the logarithmic density curvature of ρ(x). The Einstein tensor \(G_{\mu\nu}\) represents the trace-free component of SCG’s curvature tensor \(U_{ij}\), and the stress–energy tensor \(T_{\mu\nu}\) is replaced by the field’s own self-curvature \(K[\rho]\).
From this mapping Hallman derives the general identity $$ G_{\mu\nu} \;\longleftrightarrow\; -\frac{1}{\rho}\left( \frac{\partial^{2}\ln\rho}{\partial x_{\mu}\partial x_{\nu}} - \tfrac{1}{2}g_{\mu\nu}\nabla^{2}\ln\rho \right), $$ showing that Einstein’s field equations are an emergent symmetry of SCG when causal gradients are slow and isotropic. Under high curvature, the symmetry breaks and SCG predicts measurable deviations from GR—particularly in galactic rotation and strong lensing regimes—precisely where dark-matter terms are normally invoked.
The practical implication is profound: GR remains perfectly valid within its regime, but its geometric constants (G, Λ) are phenomenological encodings of deeper curvature ratios in ρ(x). As curvature gradients vanish, $$ \nabla\ln\rho \to 0 \quad\Rightarrow\quad ds^{2}\to c^{2}dt^{2}-d\mathbf{x}^{2}, $$ and SCG reduces exactly to Minkowski space. Metric physics is thus not discarded but recovered as the smooth limit of causal geometry.
Hallman concludes:
“Einstein described the shadow of curvature; SCG reveals the substance casting it. Spacetime metrics are how causal geometry looks when we stop seeing the field itself.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15062038
This paper applies SCG to a well-known observational puzzle—the constant, sunward acceleration experienced by the Pioneer 10 and 11 spacecraft—which defied explanation under conventional mechanics. Rather than invoking new forces or thermal recoil, Hallman shows that the anomaly arises naturally from the curvature gradient of the solar causal field when expressed in the SCG formalism.
In SCG the local acceleration of any freely moving object is $$ a_{\text{SCG}} = c^{2}\nabla\ln\rho, $$ and for a spherically symmetric density distribution $$ \rho(r) = \rho_{0}\!\left(\frac{r}{r_{0}}\right)^{-B}, $$ this yields a curvature-induced drift $$ a_{r} = -\,\frac{B\,c^{2}}{r}. $$ At planetary distances the Newtonian gravitational term dominates, but in the outer solar system—where curvature gradients flatten—the residual term approaches a constant value: $$ a_{\text{Pioneer}}\;\approx\;c^{2}\frac{\Delta B}{r_{\text{crit}}}, $$ with \( \Delta B \) representing the mismatch between the local and asymptotic curvature exponents.
Using the measured trajectories of Pioneer 10 and 11, Hallman determines \( \Delta B \approx 1.3\times10^{-10} \) at \( r_{\text{crit}}\!\sim\!20\text{–}70\ \text{AU} \), producing a sunward acceleration $$ a_{\text{SCG}} \approx 8.7\times10^{-10}\ {\rm m\,s^{-2}}, $$ matching the observed value without adjustable parameters. This acceleration is not a force but a geometric correction: beyond the planetary domain, the causal density gradient of the heliosphere no longer follows the pure \(r^{-2}\) dependence assumed in inverse-square law dynamics.
The analysis shows that as the heliocentric field transitions from bound to interstellar curvature, the causal geometry relaxes, slightly contracting the information gradient through which the spacecraft propagate. Signals traveling along that gradient accumulate a small additional redshift, and when interpreted through metric mechanics, this appears as a steady deceleration. In SCG terms, it is the geometry—not the spacecraft—that is changing.
This reinterpretation resolves both the magnitude and sign of the anomaly, and it predicts a similar effect for all outbound probes once they cross the solar curvature boundary. Because the magnitude scales with \(c^{2}\!\nabla\!\ln\rho\), the same mechanism also anticipates frequency drifts observed in long-baseline Doppler data for New Horizons and other deep-space transmissions.
By reframing the anomaly as a curvature-gradient phenomenon, Hallman provides the first purely geometric explanation consistent with all available data. No heat recoil, no dark forces, no ephemeris revision—just the curvature structure of ρ(x).
He closes with an understated but telling remark:
“The Pioneer craft were not slowed by something mysterious in space; they revealed that space itself is not uniform even where gravity seems silent. Their anomaly is curvature whispering that geometry never truly ends.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17205296
This paper extends the application of Spatial–Causal Geometry (SCG) to the first known interstellar visitor, 1I/ʻOumuamua. Its small, sunward acceleration could not be reconciled with general relativity, solar radiation pressure, or outgassing, yet the same curvature correction that resolved the Pioneer and flyby anomalies reproduces the data precisely—without assumptions about composition or mass.
SCG describes all motion as the equilibrium response to curvature in the scalar causal-density field ρ(x), not to applied forces. Acceleration is given by $$ \vec a(x) = c^{2}\nabla\ln\rho(x). $$ For a solar-centered field with \( \rho(r)\propto r^{-\varepsilon_{r}} \), this becomes $$ a_{r} = -\,c^{2}\frac{\varepsilon_{r}}{r} = -A_{1}\,r^{-2}, \qquad A_{1}=c^{2}\varepsilon_{r}. $$ This deterministic \(r^{-2}\) term, derived without fitting, acts as a small geometric correction to general-relativistic trajectories.
Applying this one-parameter correction to the 414 astrometric measurements (from JPL Horizons, 2017 Oct–2018 Jan) collapses the residuals from χ²ν = 2.53 to ≈ 0.22—over an order-of-magnitude improvement—with no outgassing or recoil modeling. The recovered amplitude $$ A_{1}=(4.92\pm0.16)\times10^{-6}\ {\rm m\,s^{-2}} $$ matches the empirical value adopted by Nature (Micheli et al., 2018) yet arises here directly from curvature geometry. Residuals become Gaussian and symmetric about perihelion, confirming that the anomaly reflects the field itself rather than stochastic propulsion.
In this interpretation, ʻOumuamua traversed a region where the solar causal-density gradient relaxed from its bound heliocentric form toward the interstellar background. The gradual change in ∇ln ρ introduced a small outward curvature term, producing an apparent acceleration that general relativity—lacking causal-density coupling—cannot capture. No composition-dependent physics is required; any body crossing the same gradient would display the same behavior.
The fit’s success reinforces the continuity of SCG corrections across scale: the same expression \(a_{\text{SCG}}=c^{2}\nabla\ln\rho\) that governs planetary flybys and the Pioneer trajectory also governs interstellar transits. The correction’s universality suggests that “non-gravitational” accelerations are not exceptions to gravity but confirmations of geometry.
Hallman notes that under SCG, ʻOumuamua’s behavior provides the first direct test of inter-regional causal-density variation— a condition unique to interstellar bodies. Future objects of similar trajectory should exhibit comparable deviations, offering a falsifiable prediction for upcoming surveys such as LSST and Vera Rubin.
He closes by framing the anomaly not as an astrophysical curiosity but as empirical validation of SCG’s central claim:
“Oumuamua was not pushed by hidden forces; it simply moved where geometry told it to go.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15686205
Building on the Pioneer anomaly analysis, this paper applies SCG to a second long-standing puzzle: the small, direction-dependent velocity shifts observed during planetary gravity assists, known as the Earth flyby anomaly. While conventional models attribute these deviations to navigation error or incomplete atmospheric modeling, SCG shows that they are geometric interference effects produced by the causal-density gradients of the rotating Earth itself.
In SCG the Earth’s causal field can be approximated by $$ \rho(r,\theta)=\rho_{0}\!\left(\frac{r}{r_{0}}\right)^{-B}\!\!\bigl[1+\epsilon\,P_{2}(\cos\theta)\bigr], $$ where \(P_{2}\) is the second Legendre polynomial and \(\epsilon\) encodes the planet’s equatorial flattening. The resulting curvature tensor $$ U_{ij}=-\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\ln\rho $$ contains cross-terms coupling radial and polar gradients. These cross-terms act as geometric “phase plates” that rotate the local causal gradient as the spacecraft passes from northern to southern hemispheres, producing a measurable net energy change after the encounter.
The predicted fractional velocity change is $$ \frac{\Delta v}{v} \;=\; \frac{2\,\Delta B_{\text{eq}}\,c^{2}}{v_{\infty}^{2}}\, \sin(\phi_{\text{in}}+\phi_{\text{out}})\,\sin i, $$ where \( \phi_{\text{in,out}} \) are the incoming and outgoing azimuthal angles, \(i\) is the inclination of the trajectory with respect to the equator, and \( \Delta B_{\text{eq}} \) quantifies the curvature-gradient asymmetry across the equatorial plane. This purely geometric formula reproduces both the sign and magnitude of observed anomalies for Galileo, NEAR, Rosetta, and Cassini flybys with no empirical tuning.
In SCG terms, the “energy gain” or “loss” is not mechanical—it arises from interference between the spacecraft’s information trajectory and the rotating causal gradient of the Earth. When the inbound and outbound curvature phases are misaligned, the craft exits the encounter with a slight residual offset in path coherence, manifesting as a velocity error in conventional orbital analysis.
Hallman applies the model retroactively to all documented Earth flybys, finding a consistent scaling with inclination and right ascension. The same relation also predicts that outbound trajectories aligned with the equatorial causal plane experience near-zero anomaly, matching the Voyager and Juno data.
Most notably, the paper issues a prospective prediction: the ESA JUICE mission’s 2026 Earth gravity assist is expected to display a curvature-induced Doppler residual of approximately $$ \Delta v_{\text{JUICE}} \approx +1.7 \pm 0.3\ \text{mm s}^{-1}, $$ corresponding to a fractional frequency drift of $$ \frac{\Delta f}{f} \approx 5.7\times10^{-12}, $$ detectable with standard DSN and ESTRACK precision. Because JUICE’s trajectory is highly inclined (~23°) and crosses the equatorial causal gradient near local dawn, it offers an ideal geometry to confirm or refute SCG’s prediction.
If verified, this result would mark the first deliberate, prospective confirmation of causal-geometry interference within the solar system. Hallman notes that similar signatures should appear in upcoming lunar-assist missions and in deep-space optical time-transfer experiments, where curvature phase misalignment accumulates over long baselines.
He concludes:
“The flyby anomalies were never errors in navigation; they were geometry reminding us that Earth’s curvature field is not symmetric in time. JUICE will not just fly past a planet—it will pass through the gradient that binds the planet to causality itself.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15165080
This paper demonstrates that the classical Euler equations of fluid motion are not independent laws of mechanics but the macroscopic projection of Spatial–Causal Geometry (SCG) when curvature gradients are weak. By expanding the causal field equation to first order in ∇ln ρ, Hallman shows that pressure, inertia, and vorticity all emerge from curvature continuity in the scalar field ρ(x). Fluid dynamics, in this view, is geometry written in coarse grain.
The analysis begins with the general SCG field relation $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ and expands it for small departures from equilibrium: $$ \rho=\rho_{0}(1+\delta),\qquad |\delta|\ll1. $$ To first order, the divergence term becomes $$ \nabla\!\cdot(\rho\nabla\ln\rho) \approx\nabla\!\cdot(\rho_{0}\nabla\delta) =\rho_{0}\,\nabla^{2}\delta, $$ yielding a wave equation $$ \nabla^{2}\delta=\frac{1}{c^{2}}\frac{\partial^{2}\delta}{\partial t^{2}}, $$ where temporal dependence arises as the causal-density perturbation propagates through curvature. This establishes the causal sound speed \(c\) as the limiting velocity of information flow—identical in form to acoustic propagation.
Introducing a convective derivative along a streamline, $$ \frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\!\cdot\!\nabla, $$ and applying SCG’s acceleration law \( \mathbf{a}=c^{2}\nabla\ln\rho \), Hallman recovers the Euler momentum equation in geometric form: $$ \rho\,\frac{D\mathbf{v}}{Dt}=-\,c^{2}\,\nabla\rho. $$ Identifying \(p=c^{2}\rho\) gives the standard relation $$ \rho\frac{D\mathbf{v}}{Dt}=-\,\nabla p, $$ revealing that what we call “pressure gradients” are simply manifestations of the same causal-density curvature that drives gravitational and electromagnetic motion.
Incompressible flow corresponds to the regime where ∇·(ρv)=0, i.e. causal flux is locally divergence-free. The vorticity equation follows immediately from the curl of the acceleration field: $$ \frac{D\boldsymbol{\omega}}{Dt}=(\boldsymbol{\omega}\!\cdot\!\nabla)\mathbf{v} -(\nabla\!\cdot\!\mathbf{v})\boldsymbol{\omega}, \qquad \boldsymbol{\omega}=\nabla\times\mathbf{v}, $$ matching the classical Euler formulation. Thus, the entire structure of ideal fluid mechanics is embedded within SCG geometry.
One of the most important insights of this derivation is that fluid motion inherits the same invariants seen across SCG domains: $$ \gamma_{\text{cause}}\approx1.216, \qquad \alpha_{\text{photon}}\approx0.164. $$ These constants fix the ratio between transverse curvature and longitudinal flow, defining the causal “texture” of a medium. Vorticity, wave propagation, and even shock formation all correspond to different degrees of curvature coherence within the same scalar field.
Hallman also shows that adding a small dissipative term to the curvature equation, representing non-ideal causal diffusion, $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho]+\nu\nabla^{2}\ln\rho, $$ naturally yields the Navier–Stokes equation, with kinematic viscosity $$ \nu=c^{2}\tau, $$ where τ is the characteristic causal relaxation time of the medium. Viscosity, therefore, is not a separate physical parameter but a geometric property describing how quickly curvature gradients smooth toward equilibrium.
This geometric derivation unifies continuum mechanics with the rest of SCG. Pressure, flow, and vorticity become modes of curvature propagation, and the classical Euler laws emerge as the linearized shadow of the causal field. Hallman summarizes:
“Fluids move as geometry moves—every whirl, every wave, every flow is curvature seeking rest. Euler’s equations are simply the field speaking in slow motion.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15708314
Hilbert’s Sixth Problem called for an axiomatization of physics—a single mathematical framework in which the laws of motion, probability, and field theory arise from consistent principles. This paper answers that call directly: Spatial–Causal Geometry (SCG) unifies all domains of motion under one scalar field ρ(x), where curvature—not force or probability—governs behavior.
Hallman begins by stating the central axiom:
“All measurable dynamics are expressions of curvature in a continuous scalar causal-density field ρ(x).”From this axiom follows a complete hierarchy of derived laws. Acceleration, energy, and momentum become geometric derivatives of ln ρ: $$ \mathbf{a} = c^{2}\nabla\ln\rho, \qquad E = c^{2}\rho, \qquad \mathbf{p} = \rho\,\nabla\ln\rho. $$ Each is deterministic and local, satisfying Hilbert’s requirement that physical law be built from the geometry of quantities, not statistical assumptions.
The master field equation $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho] $$ serves as the unifying postulate. Its linearized limits yield the familiar equations of physics:
In this reconstruction, probability densities are replaced by causal densities—direct measures of spatial curvature potential. Statistical mechanics emerges as the thermodynamic limit of geometric self-organization: entropy corresponds to curvature dispersion, and temperature to curvature gradient amplitude. The Boltzmann factor \( e^{-E/kT} \) becomes an approximation to the causal-density ratio \( \rho/\rho_{0}=e^{-\Delta\ln\rho} \).
Hilbert’s Sixth also demanded reconciliation between discrete and continuous description. SCG meets that requirement through its topological quantization rule: $$ \oint\nabla\ln\rho\cdot d\mathbf{s}=2\pi n, $$ where \( n\in\mathbb{Z}/2 \) generates both integer and half-integer spin states. Discrete spectra arise as stable closure modes of continuous curvature, removing the historic divide between quantum and classical behavior.
A striking feature of this synthesis is that the same geometric invariants found in optical, atomic, and cosmological contexts—particularly $$ \gamma_{\text{cause}}\approx1.216, \qquad \alpha_{\text{photon}}\approx0.164 $$ —appear again here as dimensionless ratios linking energy, curvature, and information density. They become the universal coefficients that balance continuity and discreteness in all domains, providing the quantitative backbone that Hilbert’s vision lacked.
The closing section formalizes SCG as a complete mathematical system:
Hallman closes with the statement that defines the paper’s place in mathematical physics:
“Hilbert asked that physics be axiomatized. SCG answers: geometry is the axiom. The constants we call ‘physical’ are the invariants of curvature itself.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17399688
This paper redefines the basis of quantum computation within the deterministic framework of Spatial–Causal Geometry (SCG), replacing probabilistic wavefunctions with continuous causal curvature states in the scalar field \( \rho(x) \). It demonstrates that all quantum operations—superposition, entanglement, and measurement—arise from the geometric relationships between curvature modes rather than from abstract Hilbert vectors.
Hallman introduces the concept of a curvature bit (C-bit), a bounded region of the causal field whose internal curvature oscillation encodes a reversible logical state. The superposition of two C-bit states corresponds to overlapping curvature geometries, not simultaneous probabilities. The state vector of a system is represented by the composite curvature function: $$ \rho_{\text{system}}(x) = \sum_{i=1}^{N} a_{i}\rho_{i}(x), $$ where each \( \rho_{i}(x) \) is a normalized curvature mode and the coefficients \( a_{i} \) describe geometric weighting rather than quantum amplitude.
Logical operations in this model are deterministic curvature couplings. A unitary gate becomes a reversible transformation that preserves the total curvature energy: $$ K[\rho] = c^{2}\rho(\nabla\ln\rho)^{2} = \text{constant}. $$ Quantum interference is reinterpreted as phase alignment between intersecting causal gradients. Where conventional quantum computing manipulates abstract probability amplitudes, SCG manipulates real curvature geometries that evolve predictably under the master equation: $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho]. $$
Entanglement emerges as a topological constraint linking two regions of the causal field. Instead of nonlocal correlations, the paired regions share a continuous curvature manifold; their measured outcomes correspond because both sample the same underlying geometry. Measurement therefore represents a local curvature collapse—a transition from distributed coherence to a single equilibrium configuration—without randomness or discontinuity.
Hallman shows that logical reversibility and coherence retention in quantum circuits correspond to the causal continuity condition $$ \nabla\!\cdot J_{\rho} = 0, $$ ensuring that no causal flux is lost or gained during computation. Decoherence arises only when this condition is violated—when external perturbations distort the curvature continuity between interacting C-bits.
The model reproduces the operational features of quantum algorithms while eliminating indeterminacy. Classical logic appears as the limit of large curvature separations between C-bits, while quantum logic emerges when those separations approach the coherence length defined by \( \gamma_{\text{cause}} = 1.216 \). The same invariant ratio that governs light and superconductivity thus also defines the curvature bandwidth required for stable information exchange at the causal limit.
This geometric reformulation unites computation, information, and physics: the act of computing becomes a controlled evolution of curvature within ρ(x), and the outcome of measurement is the deterministic relaxation of that geometry. SCG thereby provides a framework for physically realizable quantum computing without probabilistic collapse or hidden variables.
Hallman concludes:
“Computation is geometry in motion. A quantum gate is not a mystery—it is curvature remembering how to turn.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.17479319
This paper isolates and derives the dimensionless ratios that recur throughout the SCG framework, showing that the so-called “physical constants” are emergent geometric invariants rather than empirical inserts. By treating causal continuity as a constraint on all oscillatory and rotational solutions of $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ Hallman uncovers two universal ratios: $$ \gamma_{\text{cause}}\approx1.216,\qquad \alpha_{\text{photon}}\approx0.164. $$ These arise from the arc-length geometry of curvature closure and fix the proportionality between transverse and longitudinal scales in every bounded mode of the causal field.
The derivation begins by examining the general oscillatory solution for a bounded curvature wave, $$ L=\int_{0}^{\lambda}\!\sqrt{1+\!\left(\frac{dE}{dx}\right)^{2}}\!dx, $$ and imposing the requirement that the field return to causal equilibrium after one full cycle, \(L/\lambda=\gamma_{\text{cause}}\). Solving this closure condition yields the invariant ratio 1.216, independent of wavelength or amplitude. The photon-radius constant then follows from transverse coherence: $$ r_{\text{photon}}=\alpha_{\text{photon}}\lambda, $$ the unique value that preserves the same causal arc length across all frequencies.
Once defined, these invariants appear spontaneously in every domain of SCG:
Hallman derives a compact scaling identity linking the two: $$ \alpha_{\text{photon}}\gamma_{\text{cause}} = \frac{r_{\text{coherence}}}{\lambda_{\text{system}}} \approx 0.199, $$ showing that all SCG phenomena obey a fixed ratio of transverse to longitudinal coherence. This product appears in the curvature energy expression $$ E_{\text{SCG}}=\left(\frac{\gamma_{\text{cause}}}{2\pi\alpha_{\text{photon}}}\right)\!\frac{hc}{\lambda}, $$ yielding the ≈ 1.18 × Planck energy relation found in the photon-structure paper.
A key philosophical point follows: because these constants emerge from the curvature equation itself, they belong to geometry, not to measurement. Their numerical values encode the inherent curvature “budget” of reality—the fraction of each causal wavelength that manifests as transverse structure. SCG thus fulfills the dream of non-arbitrary constants: ratios that could not have been otherwise in any self-consistent universe.
Hallman closes with a simple but weighty observation:
“The constants are not stitched onto physics—they are the seams of geometry itself. Every ratio we measure is geometry remembering how it stays whole.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.15042913
This work asks whether the numerical constants that appear throughout physics are truly fundamental, or if they emerge from geometry itself. Using the governing relation of Spatial–Causal Geometry (SCG), $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ Hallman shows that the constants \(G,\,h,\,c,\,k_{B},\,\epsilon_{0}\) and even the empirical acceleration \(a_{0}\) are dimensional projections of a single scalar field ρ(x). They arise when curvature, energy, and information are expressed in human units, not as independent quantities written into the laws of nature.
Starting from the causal-energy identity $$ E_{\text{SCG}} = c^{2}\rho\,V, $$ the speed of light \(c\) becomes the limiting velocity of curvature propagation, not an inserted constant but a conversion between spatial and causal curvature. The gravitational constant \(G\) appears when curvature is expressed in terms of an equivalent mass density: $$ G = \frac{1}{4\pi}\,\frac{c^{2}}{\rho_{\text{cosmic}}}, $$ tying the strength of gravity to the mean curvature density of the universe. Planck’s constant \(h\) arises from the discrete closure of curvature flux in bounded systems, $$ h = \oint p\,dx = c^{2}\rho\,\lambda^{2}, $$ making quantization an artifact of geometric periodicity.
Thermodynamic and electromagnetic constants follow naturally. The Boltzmann constant \(k_{B}\) emerges from curvature dispersion: $$ k_{B}T = c^{2}\Delta\rho, $$ where \(T\) measures curvature-gradient amplitude. The permittivity and permeability of free space, \( \epsilon_{0} \) and \( \mu_{0} \), describe the ratio of transverse to longitudinal curvature in electromagnetic modes: $$ \epsilon_{0}\mu_{0} = \frac{1}{c^{2}}, $$ a direct statement of geometric coherence.
An especially significant result is the recovery of the MOND-like acceleration constant \(a_{0}\): $$ a_{0}=c^{2}\nabla\ln\rho_{\text{cosmic}}\approx1.2\times10^{-10}\ {\rm m\,s^{-2}}, $$ identical to the empirical value found in galaxy rotation data. This demonstrates that the apparent universal threshold acceleration is simply the curvature gradient of the background causal field.
Taken together, these derivations re-cast the constants as dimensional echoes of the same scalar geometry. They are conversion factors translating between curvature, energy, and information, not arbitrary knobs of the universe. Hallman summarizes:
“The constants of physics are what geometry looks like when it is written in numbers. They are not imposed—they are what remain when curvature forgets its units.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.16741464
This foundational paper establishes Spatial–Causal Geometry (SCG) as a fully specified deterministic field theory. It defines the scalar causal-density field ρ(x), derives its governing equations, and shows how all known interactions—mechanical, electromagnetic, quantum, and gravitational—emerge from a single geometric source. Where existing physics begins with empirical laws and constants, SCG begins with geometry alone.
The theory is anchored on a single postulate:
“All motion and interaction arise from curvature in the scalar field ρ(x), whose logarithmic gradient defines causal acceleration.”From this, the universal equation of motion follows directly: $$ \mathbf{a}=c^{2}\nabla\ln\rho. $$ Acceleration, energy, and potential are not independent entities but different expressions of curvature in ln ρ.
The field itself satisfies a second-order continuity equation derived from the requirement that causal flux be divergence-free: $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ where \(K[\rho]\) represents curvature energy density. This replaces both Einstein’s and Maxwell’s field equations with a single geometric law. The corresponding flux vector, $$ J_{\rho}=\rho\,\frac{\nabla\rho}{|\nabla\rho|}\,c, $$ acts as the causal current; its divergence and curl describe local stability and rotational behavior.
From these definitions arise all known dynamical relations:
Hallman introduces the causal curvature tensor, $$ U_{ij}=-\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\ln\rho, $$ which replaces the Riemann and electromagnetic tensors as the fundamental operator of curvature. All measurable quantities—force, field, energy, and momentum—are contractions of \(U_{ij}\). In particular, $$ E=c^{2}\rho,\qquad \mathbf{p}=\rho\nabla\ln\rho,\qquad \mathbf{F}=\rho\,c^{2}\nabla\ln\rho, $$ establishing a purely geometric definition of energy and momentum.
Determinism in SCG arises because ρ(x) evolves only through spatial gradients; no stochastic terms, time parameters, or wavefunction probabilities are required. Temporal evolution appears as an ordering of causal transitions rather than an external coordinate. Time, in this view, is the rate of curvature change experienced by ρ(x). This formulation satisfies Hilbert’s demand for a non-probabilistic foundation of physical law.
A key result of this formalization is that the energy density of any region can be written as $$ K[\rho]=c^{2}\rho\bigl(\nabla\ln\rho\bigr)^{2}, $$ linking geometry, energy, and information through one scalar measure. Integrating this expression over a bounded domain produces a conserved quantity that corresponds to total energy, while its gradient drives the dynamical equations of motion.
Hallman also defines the causal continuity condition, $$ \nabla\!\cdot J_{\rho}=0, $$ which guarantees that information (causal flux) cannot begin or end arbitrarily. This replaces conservation laws for mass, charge, and probability with a single principle: curvature continuity. From it, all conservation rules emerge automatically as projections along specific directions of ρ(x).
The paper closes with the geometric interpretation that unites the corpus:
“Space, energy, and motion are not separate ingredients—they are one curvature field seen from three perspectives. When we write ρ(x), we are writing the universe in its own language.”
D. J. Hallman (2025) — Zenodo DOI 10.5281/zenodo.14997137
This synthesis paper demonstrates how every major domain of physical law—mechanics, electromagnetism, quantum behavior, and cosmology—arises from a single geometric equation for the scalar causal-density field ρ(x). It serves as the first full demonstration that Spatial–Causal Geometry (SCG) reproduces all known conservation and field relations without additional postulates or constants.
Beginning from the governing law $$ \nabla\!\cdot(\rho\nabla\ln\rho)=\frac{1}{c^{2}}K[\rho], $$ Hallman expands it along orthogonal curvature modes. Each mode corresponds to a familiar “force” or interaction:
Energy and momentum follow from geometric contractions of the curvature tensor: $$ E=c^{2}\rho, \qquad \mathbf{p}=\rho\,\nabla\ln\rho, $$ and the local continuity law \( \nabla\!\cdot J_{\rho}=0 \) subsumes all conventional conservation principles. Charge conservation becomes antisymmetric curvature closure; momentum conservation, translational invariance of ρ(x); and energy conservation, invariance under curvature transport.
Hallman then demonstrates that common field equations re-emerge as limiting forms:
The unification also recovers the traditional constants as geometric factors. The light speed c appears as the maximum causal-gradient propagation rate; Planck’s constant h as the discrete curvature flux in a bounded loop; and the gravitational constant G as the scale factor linking local curvature to the background cosmic ρ. All constants are thus re-interpreted as geometry’s bookkeeping for its own self-similarity.
A concise demonstration of cross-domain prediction closes the paper. When the same curvature law is applied to atomic, astrophysical, and cosmological scales, the fitted exponents B and derived accelerations a₀ line up within 1 %. This invariance proves that SCG’s scalar field describes motion continuously across thirty orders of magnitude in scale.
Hallman’s conclusion encapsulates the theme of the entire corpus:
“Physics is not a patchwork of forces—it is one field in different postures. Every equation we use is a different way of spelling ρ(x).”
Additional insight: When viewed in the context of the full SCG corpus, this synthesis paper takes on a deeper role: it defines the symmetry operations of the causal field itself. The scalar ρ(x) supports four orthogonal projection modes—radial, transverse, rotational, and global— which later manifest respectively as gravitation, electromagnetism, quantum quantization, and cosmological curvature. This fourfold symmetry makes SCG self-contained: it can generate every dynamical law observed without appealing to external entities or dual fields. The emergence of the invariant ratio \( \gamma_{\text{cause}}\approx1.216 \) across these modes is not coincidence—it is the numerical footprint of this internal symmetry.